Lecture 9

Spatio-temporal models

Sara Martino

Dept. of Mathematical Science, NTNU

Janine Illian

University of Glasgow

Jafet Belmont

University of Glasgow

Motivation

  • Many real-world data sets vary in both space and time.

    • Examples: rainfall, temperature, air pollution, crop yield, disease spread.
  • We often want to model how similarity (correlation) changes across both dimensions.

The good news 👍

  • the framework works with spatio-temporal data as well

The bad news 👎

  • fully spatio-temporal models are complex

  • model fitting can take a long time

  • different types of spatio-temporal data structures lead to different types of complexities

Complexity of spatio-temporal models

  • additional dependencies: in space and time

  • spatial and temporal behaviour can be independent on each other, or dependent: i.e. properties of the spatial structure vary through time (or vice versa)

Several options are available in inlabru

  • Areal Model

  • Geostatistical and Point Process models

Spatio temporal models for areal data

\(\frac{\text{Number of cases } Y_{st}}{\text{Expected numer of cases }E_{st}}\) in Ohio from 1968 to 1988

Space

Spatio temporal models for areal data

\(\frac{\text{Number of cases } Y_{st}}{\text{Expected numer of cases }E_{st}}\) in Ohio from 1968 to 1988

Time

Spatio temporal models for areal data

\(\frac{\text{Number of cases } Y_{st}}{\text{Expected numer of cases }E_{st}}\) in Ohio from 1968 to 1988

Time

Spatio temporal models for areal data

The observation model \[ Y_{st}|\lambda_{st}\sim\text{Poisson}(E_{st}\lambda_{st}) \]

We are going to see different models for the linear predictor \(\eta_{st} = \log \lambda_{st}\)